Table 2.
Comparison of optimization results obtained from unimodal and multimodal test functions.
| F | D | Metrics | SA (1983) |
CGA (2000) |
GWO (2014) |
MFO (2015) |
WOA (2016) |
LMFO (2016) |
WCMFO (2019) |
ChOA (2020) |
AOA (2021) |
SMFO (2021) |
I-MFO |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| F1 | 30 | Avg | 2.251 × 1010 | 3.794 × 1010 | 8.223 × 108 | 6.952 × 109 | 1.906 × 106 | 2.402 × 107 | 1.328 × 104 | 2.238 × 1010 | 3.943 × 1010 | 3.091 × 1010 | 5.859 × 103 |
| Min | 1.897 × 1010 | 2.437 × 1010 | 4.404 × 107 | 1.027 × 109 | 5.654 × 105 | 1.731 × 107 | 1.214 × 102 | 1.123 × 1010 | 2.791 × 1010 | 2.010 × 1010 | 1.488 × 102 | ||
| 50 | Avg | 7.170 × 1010 | 1.126 × 1011 | 4.522 × 109 | 3.099 × 1010 | 7.172 × 106 | 1.091 × 108 | 2.826 × 104 | 4.407 × 1010 | 9.968 × 1010 | 6.933 × 1010 | 1.430 × 104 | |
| Min | 6.279 × 1010 | 9.612 × 1010 | 1.231 × 109 | 7.095 × 109 | 1.980 × 106 | 7.284 × 107 | 6.883 × 102 | 3.201 × 1010 | 8.424 × 1010 | 4.945 × 1010 | 1.948 × 102 | ||
| 100 | Avg | 2.830 × 1011 | 3.547 × 1011 | 3.207 × 1010 | 1.173 × 1011 | 3.677 × 107 | 7.525 × 108 | 2.017 × 105 | 1.457 × 1011 | 2.617 × 1011 | 1.908 × 1011 | 2.881 × 104 | |
| Min | 2.638 × 1011 | 3.147 × 1011 | 1.634 × 1010 | 6.748 × 1010 | 1.409 × 107 | 6.332 × 108 | 1.093 × 104 | 1.282 × 1011 | 2.343 × 1011 | 1.527 × 1011 | 1.033 × 102 | ||
| F3 | 30 | Avg | 1.439 × 105 | 1.095 × 105 | 2.993 × 104 | 1.009 × 105 | 1.715 × 105 | 2.786 × 103 | 1.887 × 103 | 5.221 × 104 | 7.488 × 104 | 8.189 × 104 | 6.106 × 102 |
| Min | 1.029 × 105 | 9.312 × 104 | 1.576 × 104 | 1.920 × 103 | 8.481 × 104 | 1.424 × 103 | 3.092 × 102 | 3.954 × 104 | 5.445 × 104 | 7.186 × 104 | 3.388 × 102 | ||
| 50 | Avg | 2.994 × 105 | 2.241 × 105 | 7.147 × 104 | 1.650 × 105 | 6.180 × 104 | 3.151 × 104 | 1.150 × 104 | 1.306 × 105 | 1.648 × 105 | 1.775 × 105 | 1.460 × 104 | |
| Min | 2.647 × 105 | 1.648 × 105 | 3.628 × 104 | 1.176 × 104 | 3.098 × 104 | 2.291 × 104 | 7.428 × 102 | 1.006 × 105 | 1.249 × 105 | 1.273 × 105 | 7.827 × 103 | ||
| 100 | Avg | 8.110 × 105 | 5.742 × 105 | 2.023 × 105 | 4.556 × 105 | 5.928 × 105 | 2.495 × 105 | 7.361 × 104 | 3.071 × 105 | 3.330 × 105 | 3.366 × 105 | 7.625 × 104 | |
| Min | 7.298 × 105 | 4.725 × 105 | 1.595 × 105 | 1.191 × 105 | 3.355 × 105 | 1.819 × 105 | 3.430 × 104 | 2.701 × 105 | 3.027 × 105 | 3.182 × 105 | 6.767 × 104 | ||
| F4 | 30 | Avg | 1.587 × 103 | 7.327 × 103 | 5.441 × 102 | 9.082 × 102 | 5.476 × 102 | 4.928 × 102 | 4.886 × 102 | 2.971 × 103 | 8.808 × 103 | 5.977 × 103 | 4.868 × 102 |
| Min | 1.354 × 103 | 6.137 × 103 | 4.963 × 102 | 5.424 × 102 | 4.995 × 102 | 4.755 × 102 | 4.239 × 102 | 1.134 × 103 | 3.824 × 103 | 3.030 × 103 | 4.704 × 102 | ||
| 50 | Avg | 7.832 × 103 | 2.608 × 104 | 8.767 × 102 | 4.097 × 103 | 6.676 × 102 | 5.907 × 102 | 5.493 × 102 | 9.176 × 103 | 2.582 × 104 | 1.879 × 104 | 5.550 × 102 | |
| Min | 6.386 × 103 | 1.598 × 104 | 6.745 × 102 | 1.216 × 103 | 5.138 × 102 | 5.084 × 102 | 4.849 × 102 | 5.017 × 103 | 1.686 × 104 | 1.005 × 104 | 4.210 × 102 | ||
| 100 | Avg | 4.778 × 104 | 1.033 × 105 | 2.812 × 103 | 2.348 × 104 | 9.992 × 102 | 7.215 × 102 | 6.423 × 102 | 2.760 × 104 | 7.703 × 104 | 5.544 × 104 | 6.318 × 102 | |
| Min | 3.845 × 104 | 8.305 × 104 | 1.870 × 103 | 6.742 × 103 | 8.615 × 102 | 6.726 × 102 | 5.980 × 102 | 2.116 × 104 | 6.019 × 104 | 3.485 × 104 | 5.772 × 102 | ||
| F5 | 30 | Avg | 8.120 × 102 | 8.777 × 102 | 5.855 × 102 | 6.894 × 102 | 8.044 × 102 | 6.278 × 102 | 6.744 × 102 | 7.877 × 102 | 7.905 × 102 | 8.721 × 102 | 5.499 × 102 |
| Min | 7.865 × 102 | 8.259 × 102 | 5.508 × 102 | 6.280 × 102 | 7.242 × 102 | 5.707 × 102 | 6.104 × 102 | 7.471 × 102 | 7.217 × 102 | 8.041 × 102 | 5.308 × 102 | ||
| 50 | Avg | 1.167 × 103 | 1.273 × 103 | 6.892 × 102 | 8.934 × 102 | 9.209 × 102 | 8.152 × 102 | 8.940 × 102 | 1.037 × 103 | 1.078 × 103 | 1.118 × 103 | 6.348 × 102 | |
| Min | 1.142 × 103 | 1.184 × 103 | 6.379 × 102 | 7.731 × 102 | 8.081 × 102 | 7.287 × 102 | 7.743 × 102 | 9.853 × 102 | 9.951 × 102 | 1.053 × 103 | 5.836 × 102 | ||
| 100 | Avg | 2.180 × 103 | 2.374 × 103 | 1.058 × 103 | 1.666 × 103 | 1.413 × 103 | 1.456 × 103 | 1.726 × 103 | 1.789 × 103 | 1.955 × 103 | 1.985 × 103 | 8.613 × 102 | |
| Min | 2.098 × 103 | 2.255 × 103 | 9.864 × 102 | 1.455 × 103 | 1.329 × 103 | 1.226 × 103 | 1.328 × 103 | 1.724 × 103 | 1.842 × 103 | 1.875 × 103 | 7.885 × 102 | ||
| F6 | 30 | Avg | 6.582 × 102 | 6.734 × 102 | 6.043 × 102 | 6.267 × 102 | 6.671 × 102 | 6.038 × 102 | 6.225 × 102 | 6.604 × 102 | 6.655 × 102 | 6.830 × 102 | 6.000 × 102 |
| Min | 6.468 × 102 | 6.612 × 102 | 6.011 × 102 | 6.144 × 102 | 6.410 × 102 | 6.017 × 102 | 6.095 × 102 | 6.386 × 102 | 6.476 × 102 | 6.615 × 102 | 6.000 × 102 | ||
| 50 | Avg | 6.835 × 102 | 6.936 × 102 | 6.105 × 102 | 6.437 × 102 | 6.760 × 102 | 6.094 × 102 | 6.400 × 102 | 6.720 × 102 | 6.837 × 102 | 6.890 × 102 | 6.000 × 102 | |
| Min | 6.691 × 102 | 6.784 × 102 | 6.052 × 102 | 6.270 × 102 | 6.638 × 102 | 6.034 × 102 | 6.165 × 102 | 6.608 × 102 | 6.747 × 102 | 6.780 × 102 | 6.000 × 102 | ||
| 100 | Avg | 7.210 × 102 | 7.176 × 102 | 6.275 × 102 | 6.648 × 102 | 6.768 × 102 | 6.398 × 102 | 6.664 × 102 | 6.860 × 102 | 7.028 × 102 | 7.030 × 102 | 6.000 × 102 | |
| Min | 7.164 × 102 | 7.143 × 102 | 6.229 × 102 | 6.466 × 102 | 6.676 × 102 | 6.222 × 102 | 6.526 × 102 | 6.761 × 102 | 6.970 × 102 | 6.865 × 102 | 6.000 × 102 | ||
| F7 | 30 | Avg | 1.728 × 103 | 1.965 × 103 | 8.418 × 102 | 1.011 × 103 | 1.238 × 103 | 8.735 × 102 | 8.985 × 102 | 1.190 × 103 | 1.295 × 103 | 1.349 × 103 | 7.964 × 102 |
| Min | 1.593 × 103 | 1.798 × 103 | 7.801 × 102 | 8.671 × 102 | 1.089 × 103 | 8.438 × 102 | 8.402 × 102 | 1.063 × 103 | 1.154 × 103 | 1.175 × 103 | 7.595 × 102 | ||
| 50 | Avg | 3.572 × 103 | 3.656 × 103 | 1.015 × 103 | 1.701 × 103 | 1.684 × 103 | 1.092 × 103 | 1.141 × 103 | 1.663 × 103 | 1.860 × 103 | 1.919 × 103 | 9.208 × 102 | |
| Min | 3.305 × 103 | 3.300 × 103 | 9.654 × 102 | 1.113 × 103 | 1.500 × 103 | 1.065 × 103 | 1.020 × 103 | 1.464 × 103 | 1.744 × 103 | 1.769 × 103 | 8.168 × 102 | ||
| 100 | Avg | 1.014 × 104 | 9.052 × 103 | 1.710 × 103 | 4.169 × 103 | 3.250 × 103 | 1.736 × 103 | 1.988 × 103 | 3.320 × 103 | 3.712 × 103 | 3.891 × 103 | 1.356 × 103 | |
| Min | 9.035 × 103 | 8.338 × 103 | 1.542 × 103 | 2.576 × 103 | 2.814 × 103 | 1.649 × 103 | 1.531 × 103 | 3.127 × 103 | 3.579 × 103 | 3.536 × 103 | 1.140 × 103 | ||
| F8 | 30 | Avg | 1.116 × 103 | 1.166 × 103 | 8.713 × 102 | 9.790 × 102 | 1.000 × 103 | 9.379 × 102 | 9.841 × 102 | 1.032 × 103 | 1.041 × 103 | 1.096 × 103 | 8.574 × 102 |
| Min | 1.074 × 103 | 1.144 × 103 | 8.435 × 102 | 8.938 × 102 | 9.488 × 102 | 8.797 × 102 | 9.344 × 102 | 9.726 × 102 | 1.002 × 103 | 1.058 × 103 | 8.418 × 102 | ||
| 50 | Avg | 1.467 × 103 | 1.573 × 103 | 9.792 × 102 | 1.229 × 103 | 1.249 × 103 | 1.119 × 103 | 1.213 × 103 | 1.305 × 103 | 1.426 × 103 | 1.404 × 103 | 9.201 × 102 | |
| Min | 1.427 × 103 | 1.519 × 103 | 9.384 × 102 | 1.118 × 103 | 1.132 × 103 | 1.062 × 103 | 1.087 × 103 | 1.251 × 103 | 1.339 × 103 | 1.320 × 103 | 8.796 × 102 | ||
| 100 | Avg | 2.499 × 103 | 2.751 × 103 | 1.397 × 103 | 1.968 × 103 | 1.897 × 103 | 1.740 × 103 | 2.026 × 103 | 2.157 × 103 | 2.404 × 103 | 2.422 × 103 | 1.160 × 103 | |
| Min | 2.448 × 103 | 2.620 × 103 | 1.225 × 103 | 1.717 × 103 | 1.716 × 103 | 1.531 × 103 | 1.756 × 103 | 2.052 × 103 | 2.248 × 103 | 2.276 × 103 | 1.087 × 103 | ||
| F9 | 30 | Avg | 1.079 × 104 | 1.239 × 104 | 1.384 × 103 | 6.278 × 103 | 7.233 × 103 | 1.015 × 103 | 8.747 × 103 | 6.612 × 103 | 5.570 × 103 | 9.591 × 103 | 9.882 × 102 |
| Min | 9.017 × 103 | 8.805 × 103 | 1.025 × 103 | 4.471 × 103 | 4.425 × 103 | 9.056 × 102 | 5.118 × 103 | 4.627 × 103 | 4.101 × 103 | 7.754 × 103 | 9.065 × 102 | ||
| 50 | Avg | 3.302 × 104 | 4.335 × 104 | 4.571 × 103 | 1.644 × 104 | 1.783 × 104 | 1.306 × 103 | 2.195 × 104 | 2.591 × 104 | 2.277 × 104 | 3.066 × 104 | 1.305 × 103 | |
| Min | 2.622 × 104 | 3.603 × 104 | 2.135 × 103 | 8.748 × 103 | 1.187 × 104 | 9.481 × 102 | 1.190 × 104 | 1.969 × 104 | 1.804 × 104 | 1.925 × 104 | 9.853 × 102 | ||
| 100 | Avg | 1.197 × 105 | 1.243 × 105 | 2.638 × 104 | 4.507 × 104 | 3.820 × 104 | 1.421 × 104 | 5.208 × 104 | 6.813 × 104 | 5.374 × 104 | 6.959 × 104 | 7.081 × 103 | |
| Min | 1.034 × 105 | 1.079 × 105 | 1.102 × 104 | 3.679 × 104 | 2.557 × 104 | 3.037 × 103 | 3.986 × 104 | 5.806 × 104 | 4.673 × 104 | 5.852 × 104 | 4.038 × 103 | ||
| F10 | 30 | Avg | 7.764 × 103 | 8.149 × 103 | 3.909 × 103 | 5.130 × 103 | 6.156 × 103 | 4.422 × 103 | 4.808 × 103 | 8.037 × 103 | 6.487 × 103 | 8.363 × 103 | 2.745 × 103 |
| Min | 6.721 × 103 | 7.474 × 103 | 2.718 × 103 | 3.575 × 103 | 4.506 × 103 | 3.149 × 103 | 3.332 × 103 | 7.199 × 103 | 5.410 × 103 | 7.473 × 103 | 1.941 × 103 | ||
| 50 | Avg | 1.426 × 104 | 1.446 × 104 | 6.428 × 103 | 8.566 × 103 | 9.478 × 103 | 7.081 × 103 | 7.956 × 103 | 1.419 × 104 | 1.223 × 104 | 1.368 × 104 | 4.799 × 103 | |
| Min | 1.337 × 104 | 1.342 × 104 | 4.582 × 103 | 6.288 × 103 | 6.969 × 103 | 6.045 × 103 | 6.204 × 103 | 1.301 × 104 | 1.058 × 104 | 1.234 × 104 | 3.734 × 103 | ||
| 100 | Avg | 3.136 × 104 | 3.121 × 104 | 1.497 × 104 | 1.728 × 104 | 2.012 × 104 | 1.758 × 104 | 1.618 × 104 | 3.141 × 104 | 2.783 × 104 | 3.071 × 104 | 1.168 × 104 | |
| Min | 3.040 × 104 | 2.990 × 104 | 1.141 × 104 | 1.417 × 104 | 1.687 × 104 | 1.631 × 104 | 1.147 × 104 | 3.035 × 104 | 2.582 × 104 | 2.843 × 104 | 9.083 × 103 | ||
| Ranking | 30 | W|T|L | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 9/0/0 |
| 50 | W|T|L | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 2/0/7 | 0/0/9 | 0/0/9 | 0/0/9 | 7/0/2 | |
| 100 | W|T|L | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 0/0/9 | 1/0/8 | 0/0/9 | 0/0/9 | 0/0/9 | 8/0/1 |